Simon Thomas "Superrigidity and countable Borel equivalence relations" Corollary 4.9 gives a countable Borel equivalence relation E shows E and 2E are not Borel bireducible. (The examples involve probability measures and group actions.) It is quite possible that under $\mathsf{AD}^+$, $|\mathbb{R} / E| < |\mathbb{R} / E \sqcup \mathbb{R} / E|$ for this equivalence relation (as Woodin claims).
Woodin "The cardinals below $|[\omega_1]^{<\omega_1}|$" Corollary 71 gives a set $X_0$ so that $|X_0| < |X_0 \sqcup X_0|$ under $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$.
By Hjorth's $E_0$-dichotomy ("A Dichotomy for the Definable Universe"), under $\mathsf{AD}^+$, every set which is a surjective image of $\mathbb{R}$ either injects into the power set of an ordinal (is linearly orderable) or $\mathbb{R} / E_0$ injects into it (is not linearly orderable). Woodin's example is a set of the former type and Thomas's example is a set of the latter type.